Robot Manipulator Jacobian Velocity Calculator
Manipulator kinematics & velocity mapping: Compute the geometric Jacobian matrix, map joint velocities to Cartesian speeds ($mathbf{v} = J dot{mathbf{q}}$), and calculate the Yoshikawa manipulability index ($w$).
2-Link Planar Arm State & Joint Rates
Jacobian & Cartesian Velocities
Recommended Tools & Equipment
Tested hardware and components for high reliability
Frequently Asked Questions
What is the physical meaning of the manipulability ellipsoid?
The manipulability ellipsoid visualizes the set of all Cartesian velocities the end-effector can achieve for a unit ball of joint velocities (||q_dot|| <= 1). The principal axes are aligned with the eigenvectors of JJ^T, and their lengths equal the singular values of the Jacobian.
How do robot controllers handle singularities without crashing?
Industrial controllers employ Damped Least Squares (DLS, Levenberg-Marquardt) inversion: J_dls = J^T (J J^T + λ² I)^(-1). Near a singularity, damping factor λ² smoothly sacrifices exact Cartesian path tracking to prevent joint velocity blowups.
What is the difference between geometric and analytical Jacobians?
The geometric Jacobian maps joint rates directly to physical linear velocity v and angular velocity ω. The analytical Jacobian maps joint rates to time derivatives of a chosen set of minimal orientation representations (such as Euler angles, dX/dt = J_a · q_dot).